jee-main

Papers (169)
2025
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2024
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2023
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2022
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2021
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2020
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2019
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2018
08apr 29 15apr 28 15apr_shift1 28 15apr_shift2 2 16apr 15
2017
02apr 28 08apr 29 09apr 30
2016
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2015
04apr 29 10apr 30
2014
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2013
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
2012
07may 18 12may 22 19may 13 26may 17 offline 30
2011
jee-main_2011.pdf 13
2010
jee-main_2010.pdf 1
2009
jee-main_2009.pdf 1
2008
jee-main_2008.pdf 1
2007
jee-main_2007.pdf 38
2005
jee-main_2005.pdf 19
2004
jee-main_2004.pdf 11
2003
jee-main_2003.pdf 9
2002
jee-main_2002.pdf 8
2025 session1_23jan_shift1

25 maths questions

Q1 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
(1) - 1080
(2) - 1020
(3) - 1200
(4) - 120
Q2 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4 . The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 4 } { 9 }$
(4) $\frac { 3 } { 5 }$
Q3 Vectors 3D & Lines Section Division and Coordinate Computation View
Let the position vectors of the vertices $A , B$ and $C$ of a tetrahedron $A B C D$ be $\hat { \mathbf { i } } + 2 \hat { \mathbf { j } } + \hat { \mathbf { k } } , \hat { \mathbf { i } } + 3 \hat { \mathbf { j } } - 2 \hat { k }$ and $2 \hat { i } + \hat { j } - \hat { k }$ respectively. The altitude from the vertex $D$ to the opposite face $A B C$ meets the median line segment through $A$ of the triangle $A B C$ at the point $E$. If the length of $A D$ is $\frac { \sqrt { } \overline { 110 } } { 3 }$ and the volume of the tetrahedron is $\frac { \sqrt { 805 } } { 6 \sqrt { 2 } }$, then the position vector of $E$ is
(1) $\frac { 1 } { 12 } ( 7 \hat { \mathbf { i } } + 4 \hat { \mathbf { j } } + 3 \hat { k } )$
(2) $\frac { 1 } { 2 } ( \hat { i } + 4 \hat { j } + 7 \hat { k } )$
(3) $\frac { 1 } { 6 } ( 12 \hat { i } + 12 \hat { j } + \hat { k } )$
(4) $\frac { 1 } { 6 } ( 7 \hat { \mathrm { i } } + 12 \hat { \mathrm { j } } + \hat { \mathrm { k } } )$
Q4 Matrices Matrix Algebra and Product Properties View
If $\mathrm { A } , \mathrm { B }$, and $\left( \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) \right)$ are non-singular matrices of same order, then the inverse of $\mathrm { A } \left( \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) \right) ^ { - 1 } \mathrm {~B}$, is equal to
(1) $\mathrm { AB } ^ { - 1 } + \mathrm { A } ^ { - 1 } \mathrm {~B}$
(2) $\operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right)$
(3) $\frac { A B ^ { - 1 } } { | A | } + \frac { B A ^ { - 1 } } { | B | }$
(4) $\frac { 1 } { | A B | } ( \operatorname { adj } ( B ) + \operatorname { adj } ( A ) )$
Q5 Measures of Location and Spread View
Marks obtains by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is
(1) 52
(2) 48
(3) 44
(4) 40
Q6 Differential equations Solving Separable DEs with Initial Conditions View
Let a curve $y = f ( x )$ pass through the points $( 0,5 )$ and $\left( \log _ { e } 2 , k \right)$. If the curve satisfies the differential equation $2 ( 3 + y ) e ^ { 2 x } d x - \left( 7 + e ^ { 2 x } \right) d y = 0$, then $k$ is equal to
(1) 4
(2) 32
(3) 8
(4) 16
Q7 Sign Change & Interval Methods View
If the function $f ( x ) = \left\{ \begin{array} { l } \frac { 2 } { x } \left\{ \sin \left( k _ { 1 } + 1 \right) x + \sin \left( k _ { 2 } - 1 \right) x \right\} , \quad x < 0 \\ 4 , \quad x = 0 \end{array} \quad \right.$ is continuous at $\mathrm { x } = 0$, then $\mathrm { k } _ { 1 } ^ { 2 } + \mathrm { k } _ { 2 } ^ { 2 }$ is equal to
(1) 20
(2) 5
(3) 8
(4) 10
Q8 Straight Lines & Coordinate Geometry Line-Curve Intersection and Chord Properties View
If the line $3 x - 2 y + 12 = 0$ intersects the parabola $4 y = 3 x ^ { 2 }$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $A B$ subtends an angle equal to
(1) $\tan ^ { - 1 } \left( \frac { 4 } { 5 } \right)$
(2) $\tan ^ { - 1 } \left( \frac { 9 } { 7 } \right)$
(3) $\tan ^ { - 1 } \left( \frac { 11 } { 9 } \right)$
(4) $\frac { \pi } { 2 } - \tan ^ { - 1 } \left( \frac { 3 } { 2 } \right)$
Q9 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let $P$ be the foot of the perpendicular from the point $Q ( 10 , - 3 , - 1 )$ on the line $\frac { x - 3 } { 7 } = \frac { y - 2 } { - 1 } = \frac { z + 1 } { - 2 }$. Then the area of the right angled triangle $P Q R$, where $R$ is the point $( 3 , - 2,1 )$, is
(1) $9 \sqrt { 15 }$
(2) $\sqrt { 30 }$
(3) $8 \sqrt { 15 }$
(4) $3 \sqrt { 30 }$
Q10 Vectors Introduction & 2D Expressing a Vector as a Linear Combination View
Let the $\operatorname { arc } A C$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$, divides the arc $A C$ such that $\frac { \text { length of } \operatorname { arc } A B } { \text { length of } \operatorname { arc } B C } = \frac { 1 } { 5 }$, and $\overrightarrow { O C } = \alpha \overrightarrow { O A } + \beta \overrightarrow { O B }$, then $\alpha + \sqrt { 2 } ( \sqrt { 3 } - 1 ) \beta$ is equal to
(1) $2 \sqrt { 3 }$
(2) $2 - \sqrt { 3 }$
(3) $5 \sqrt { 3 }$
(4) $2 + \sqrt { 3 }$
Q11 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
Let $f ( x ) = \log _ { \mathrm { e } } x$ and $g ( x ) = \frac { x ^ { 4 } - 2 x ^ { 3 } + 3 x ^ { 2 } - 2 x + 2 } { 2 x ^ { 2 } - 2 x + 1 }$. Then the domain of $f \circ g$ is
(1) $[ 0 , \infty )$
(2) $[ 1 , \infty )$
(3) $( 0 , \infty )$
(4) $\mathbb { R }$
Q12 Matrices Linear System and Inverse Existence View
If the system of equations $$( \lambda - 1 ) x + ( \lambda - 4 ) y + \lambda z = 5$$ $$\lambda x + ( \lambda - 1 ) y + ( \lambda - 4 ) z = 7$$ $$( \lambda + 1 ) x + ( \lambda + 2 ) y - ( \lambda + 2 ) z = 9$$ has infinitely many solutions, then $\lambda ^ { 2 } + \lambda$ is equal to
(1) 6
(2) 10
(3) 20
(4) 12
Q13 Permutations & Arrangements Word Permutations with Repeated Letters View
The number of words, which can be formed using all the letters of the word ``DAUGHTER'', so that all the vowels never come together, is
(1) 36000
(2) 37000
(3) 34000
(4) 35000
Q14 Probability Definitions Combinatorial Counting (Non-Probability) View
Let $\mathbf { R } = \{ ( 1,2 ) , ( 2,3 ) , ( 3,3 ) \}$ be a relation defined on the set $\{ 1,2,3,4 \}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:
(1) 10
(2) 7
(3) 8
(4) 9
Q15 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Let the area of a $\triangle P Q R$ with vertices $P ( 5,4 ) , Q ( - 2,4 )$ and $R ( a , b )$ be 35 square units. If its orthocenter and centroid are $O \left( 2 , \frac { 14 } { 5 } \right)$ and $C ( c , d )$ respectively, then $c + 2 d$ is equal to
(1) $\frac { 8 } { 3 }$
(2) $\frac { 7 } { 3 }$
(3) 2
(4) 3
Q16 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The value of $\int _ { e ^ { 2 } } ^ { e ^ { 4 } } \frac { 1 } { x } \left( \frac { e ^ { \left( \left( \log _ { e } x \right) ^ { 2 } + 1 \right) ^ { - 1 } } } { e ^ { \left( \left( \log _ { e } x \right) ^ { 2 } + 1 \right) ^ { - 1 } } + e ^ { \left( \left( 6 - \log _ { e } x \right) ^ { 2 } + 1 \right) ^ { - 1 } } } \right) d x$ is
(1) 2
(2) $\log _ { e } 2$
(3) 1
(4) $e ^ { 2 }$
Q17 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
Let $\left| \frac { \bar { z } - i } { 2 \bar { z } + i } \right| = \frac { 1 } { 3 } , z \in C$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $( 0,0 ) , \mathrm { C }$ and $( \alpha , 0 )$ is 11 square units, then $\alpha ^ { 2 }$ equals:
(1) 50
(2) 100
(3) $\frac { 81 } { 25 }$
(4) $\frac { 121 } { 25 }$
Q18 Trig Proofs Trigonometric Identity Simplification View
The value of $\left( \sin 70 ^ { \circ } \right) \left( \cot 10 ^ { \circ } \cot 70 ^ { \circ } - 1 \right)$ is
(1) $2/3$
(2) 1
(3) 0
(4) $3/2$
Q19 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
Let $\mathrm { I } ( x ) = \int \frac { d x } { ( x - 11 ) ^ { \frac { 11 } { 13 } } ( x + 15 ) ^ { \frac { 15 } { 13 } } }$. If $\mathrm { I } ( 37 ) - \mathrm { I } ( 24 ) = \frac { 1 } { 4 } \left( \frac { 1 } { \mathrm {~b} ^ { \frac { 1 } { 13 } } } - \frac { 1 } { \mathrm { c } ^ { \frac { 1 } { 13 } } } \right) , \mathrm { b } , \mathrm { c } \in \mathrm { N }$, then $3 ( \mathrm {~b} + \mathrm { c } )$ is equal to
(1) 22
(2) 39
(3) 40
(4) 26
Q20 Reciprocal Trig & Identities View
If $\frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 4 }$, then $\cos ^ { - 1 } \left( \frac { 12 } { 13 } \cos x + \frac { 5 } { 13 } \sin x \right)$ is equal to
(1) $x - \tan ^ { - 1 } \frac { 4 } { 3 }$
(2) $x + \tan ^ { - 1 } \frac { 4 } { 5 }$
(3) $x - \tan ^ { - 1 } \frac { 5 } { 12 }$
(4) $x + \tan ^ { - 1 } \frac { 5 } { 12 }$
Q21 Circles Circle Equation Derivation View
Let the circle $C$ touch the line $x - y + 1 = 0$, have the centre on the positive x-axis, and cut off a chord of length $\frac { 4 } { \sqrt { 13 } }$ along the line $- 3 x + 2 y = 1$. Let H be the hyperbola $\frac { x ^ { 2 } } { \alpha ^ { 2 } } - \frac { y ^ { 2 } } { \beta ^ { 2 } } = 1$, whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of $C$. Then $2 \alpha ^ { 2 } + 3 \beta ^ { 2 }$ is equal to $\_\_\_\_$
Q22 Discriminant and conditions for roots Condition for repeated (equal/double) roots View
If the equation $\mathrm { a } ( \mathrm { b} - \mathrm { c } ) \mathrm { x } ^ { 2 } + \mathrm { b } ( \mathrm { c } - \mathrm { a } ) \mathrm { x } + \mathrm { c } ( \mathrm { a } - \mathrm { b } ) = 0$ has equal roots, where $\mathrm { a } + \mathrm { c } = 15$ and $\mathrm { b } = \frac { 36 } { 5 }$, then $a ^ { 2 } + c ^ { 2 }$ is equal to
Q23 Curve Sketching Number of Solutions / Roots via Curve Analysis View
If the set of all values of a, for which the equation $5 x ^ { 3 } - 15 x - a = 0$ has three distinct real roots, is the interval $( \alpha , \beta )$, then $\beta - 2 \alpha$ is equal to $\_\_\_\_$
Q24 Binomial Theorem (positive integer n) Count Integral or Rational Terms in a Binomial Expansion View
The sum of all rational terms in the expansion of $\left( 1 + 2 ^ { 1 / 2 } + 3 ^ { 1 / 2 } \right) ^ { 6 }$ is equal to
Q25 Areas by integration View
If the area of the larger portion bounded between the curves $x ^ { 2 } + y ^ { 2 } = 25$ and $y = | x - 1 |$ is $\frac { 1 } { 4 } ( b \pi + c ) , b , c \in N$, then $b + c$ is equal to